A Wishart kernel density estimator (KDE) is introduced for density estimation in the cone of positive definite matrices. The estimator is boundary-aware and mitigates the boundary bias suffered by conventional KDEs, while remaining simple to implement. Its mean squared error, uniform strong consistency on expanding compact sets, and asymptotic normality are established under the Lebesgue measure and suitable mixing conditions. This work represents the first study of density estimation for dependent data on this space under any metric. For independent observations, an asymptotic upper bound on the mean absolute error is also derived. A simulation study compares the performance of the Wishart KDE with that of the log-Gaussian KDE, another boundary-aware estimator based on the matrix-variate lognormal distribution proposed by Schwartzman [Int. Stat. Rev., 2016, 84(3), 456--486], and with the naive Gaussian KDE on the ambient Euclidean space. When estimating the stationary marginal density of a Wishart autoregressive process for several autoregressive coefficient matrices and innovation covariance matrices, the Wishart KDE exhibits the best overall accuracy and stability. The practical utility of the Wishart KDE is illustrated by estimating the marginal density of a one-year time series of realized covariance matrices computed from 5-minute intra-day returns on Amazon Corp. shares and on the Standard & Poor's 500 exchange-traded fund. All code is publicly available via the R package ksm to facilitate implementation of the method and reproducibility of the findings.

翻译：本文引入了一种Wishart核密度估计器，用于正定矩阵锥上的密度估计。该估计器具有边界感知特性，能够减轻传统核密度估计器所遭受的边界偏差，同时保持实现简便。在勒贝格测度和适当混合条件下，建立了其均方误差、在扩张紧集上的一致强相合性以及渐近正态性。该工作是在任意度量下对此空间上相依数据进行密度估计的首项研究。针对独立观测值，还导出了平均绝对误差的渐近上界。仿真研究比较了Wishart核密度估计器与对数高斯核密度估计器（另一种基于Schwartzman [Int. Stat. Rev., 2016, 84(3), 456--486] 提出的矩阵变量对数正态分布的边界感知估计器）以及环境欧氏空间上的朴素高斯核密度估计器的性能。在针对多个自回归系数矩阵和新息协方差矩阵估计Wishart自回归过程的平稳边际密度时，Wishart核密度估计器展现出最佳的整体精度和稳定性。通过估计亚马逊公司股票和标准普尔500交易所交易基金5分钟日内收益率计算的已实现协方差矩阵一年时间序列的边际密度，展示了Wishart核密度估计器的实际效用。所有代码均通过R包ksm公开发布，以促进方法的实施和结果的再现性。